An Erdős-Ko-Rado theorem for the derangement graph of \(PGL_3(q)\) acting on the projective plane

Abstract

In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group (PGL_3(q)), in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line. This gives the first evidence to the veracity of Conjecture~(2) from K.~Meagher, P.~Spiga, An Erdős-Ko-Rado theorem for the derangement graph of (\mathrm{PGL}(2,q)) acting on the projective line, ( \textit{J. Comb. Theory Series A} \textbf{118} ) (2011), 532--544.